jmc

By the Rational Root Theorem, the only possible rational roots are of the form $\pm \frac{a}{b},$ where $a$ divides 15 and $b$ divides 9. Thus, the possible rational roots are $$\pm 1,\pm 3,\pm 5,\pm 15,\pm \frac{1}{3},\pm \frac{5}{3},\pm \frac{1}{9},\pm \frac{5}{9}.$$Thus, there are $\boxed{16}$ possible rational roots.